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\cl{\bf              About the Koch Snowflake (or Island)}

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\cl{                               H. Karcher                         }
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    The Koch Snowflake Curve (aka the Koch Island) is a fractal planar curve of infinite length and dimensionapproximately 1.262. It is defined as the limit of a sequence of polygonal curves defined recursively as follows:\item{1)}  The first polygon is an equilateral triangle.\item{2)} The (n + 1)-st polygon is created from the\break    n-th polygon by applying the following rule to    each edge:  construct an equilateral triangle     with base the middle third of the edge and     pointing towards the outside of the polygon,    then remove the base of this new triangle.
 
 \noindentNote that at each step the number of segments increases by a factor 4 with the new segments being one third the length of the old ones. Since all end points of segments are already points on the limit curve we see that no part of the limit curve has finite length. 
 
 \vfil\ejectConsider the union of those disks which have a segment of one polygonal approximation as a diameter, then this union covers all the further approximations. From one step to the next the diameter of the disks shrinks to one thirdwhile the number of disks is multiplied by 4---so that the area of these covering disk unions converges to zero. The fractal Hausdorff d-measure is defined as the infimum(as the diameter goes to zero) of the quantity (diameter)${}^d \times$ (number-of-disks),and the fractal Hausdorff dimension is the infimum of those d for which the d-measure is 0. This shows that the Hausdorff dimension of the Koch curve is less than or equal to   $\log(4) / \log(3)$, and since the union of the disks of every second segment does not cover the limit curve one can conclude that the Hausdorff dimension is precisely $\log(4) / \log(3)$.

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